Here is (my summary, my words) the most important idea contributed to Popper's philosophy of science by someone other than Popper. It was contributed by David Deutsch in his book The Fabric of Reality:
Most ideas are criticized and rejected for being bad explanations. This is true even in science where they could be tested. Even most proposed scientific ideas are rejected, without testing, for being bad explanations.
Although tests are valuable, Popper's over-emphasis on testing mischaracterizes science and sets it further apart from philosophy than need be. In both science and abstract philosophy, most criticism revolves around good and bad explanations. It's largely the same epistemology. The possibility of empirical testing in science is a nice bonus, not a necessary part of creating knowledge.
In his book, David Deutsch gives this example: Consider the theory that eating grass cures colds. He says we can reject this theory without testing it.
He's right, isn't he? Should we hire a bunch of sick college students to eat grass? That would be silly. There is no explanation of how grass cures colds, so nothing worth testing. (Non-explanation is a common type of bad explanation!)
Narrow focus on testing -- especially as a substitute for support/justification -- is one of the major ways of misunderstanding Popperian philosophy. Deutsch's improvement shows how its importance is overrated and, besides being true, is better in keeping with the fallibilist spirit of Popper's thought (we don't need something "harder" or "more sciency" or whatever than critical argument!).
Emphasis on explanations is a theme with Deutsch. His upcoming book, The Beginning of Infinity is subtitled "Explanations that transform the world".
Another big idea of Deutsch's is that Popperian epistemology is true for all people. It sounds obvious when stated in that form, but it becomes controversial when one mentions that children are included in "all people". I think Popper would have approved of this, but he didn't go through and explain the consequences and implications for education. Deutsch has done so in detail.
In An Introduction To The Thought Of Karl Popper, p 41, Roberta Corvi summarizes Popper, "The practical problem of induction is thereby solved: it is transformed into the problem of testing a theory". This is just the kind of empiricist mistake which Deutsch has improved on. Empirical approaches are insufficient in general because they cannot address philosophy (and epistemology should apply to all knowledge), but even in science when testing is possible, strong empiricism (i.e. we learn primarily using observation) is still a mistake.